Notation and conventions. Throughout this paper we identify the interval \([0,1)\)  with the circle \(\mathbb {S}\) . We denote by \(\mathbb {E}\)  the real line \(\mathbb {R}\)  (non-periodic problem) or \(\mathbb {S}\)  (periodic case), while \(L^1(\mathbb {E})\)  corresponds to the space of the integrable functions on \(\mathbb {E}\) . The norm of a Banach space \(X\)  is denote by \(\Vert \cdot \Vert _X\) . Given \(f,\,g\in L^1(\mathbb {E})\) , their convolution is denoted by \(f\ast g\) , whereas for each \(s\in \mathbb {R}\) , the Sobolev space of order \(s\)  is denoted by \(H^s(\mathbb {E})\)  (for further details, see [1]}, [2]}). If \(u=u(t,x)\) , we use \(u_0(x)\)  to denote the function \(x\mapsto u(0,x)\) . Moreover, we also define \(m:=u-u_{xx}\) , for \(t>0\) , and \(m_0:=u_0-u_0^{\prime \prime }\) . Let \(\lfloor \cdot \rfloor \)  denote the greatest integer function and we recall the Helmholtz operator \(\Lambda ^2=1-\partial _x^2\)  and its inverse \(\Lambda ^{-2}=(1-\partial _x^2)^{-1}\) , given by \(\Lambda ^{-2}(f)=g\ast f\) , where
\(g(x)=\left\lbrace \begin{array}{ll}\displaystyle {{2}},&\text{if}\,\,\mathbb {E}=\mathbb {R},\\\\\displaystyle {\frac{\cosh (x-\lfloor x \rfloor - 1/2)}{2\sinh (1/2)}}, & \text{if}\,\,\mathbb {E}=\mathbb {S}.\end{array}\right.\) 
